Heterogeneity and Periodicity in Dynamic Optimization

Heterogeneity can play a substantial role in the evolution of populations, economic systems, epidemic diseases, physical systems, etc. While this has been well recognized in population or physical contexts, the investigation of the role of heterogeneity in economics is a modern, quickly developing area. Dynamic optimization problems for heterogeneous systems first appeared in the population dynamics literature in the eighties. The development of the respective mathematical theory in the last decade created a new powerful research instrument applicable also in economics, management and social sciences. In this proposal heterogeneous control systems are described mathematically by first order distributed controlled differential equations with non-local dynamics and endogenous side-conditions. The theoretical investigations within the present project are motivated by several problems in economics and management that cannot be addressed by applying the existing theory. One of these problems is to characterize the optimal trade-off between investments in capital accumulation versus technological growth. Here the heterogeneity arises due to the simultaneous availability of a variety of technologies and a variety of consumption goods. Both varieties depend dynamically on the investments in research and development (R&D). The main challenge here is to analyze the determinants of technological change, in particular, the role of the demographic factors for the technological advancement. This problem requires developing the theory of optimal control for heterogeneous systems for which the range of heterogeneity depends dynamically on the control policy. This comprises the first main theoretical goal of the project, which includes optimality conditions, numerical methods, and computational tools, in particular for infinite horizon problems. The project includes a second application of the theory and software developed within this goal, which is the investigation of the fundamental question in the context of climate change and global warming: Can the socially optimal abatement policy for greenhouse gases be "implemented" without any direct emission restrictions, controlling only the composition of the investments in R&D? The heterogeneity arises from the coexistence of technologies with different productivities and emission rates, hence the range of heterogeneity depends on the investment policy. A second theoretical goal of the project is to develop mathematical (analytical and numerical) instruments for the investigation of optimal cyclic heterogeneous processes. One very important motivation for this study is industrial fishing in the oceans. It has been observed that the selective industrial fishing leads to fast detrimental evolutionary changes. A model that is able to capture simultaneously the selectivity of fishing (with respect to size) and the evolutionary changes needs at least two parameters of heterogeneity: size and genetic parameters. There is a certain practical and mathematical evidence that the optimal fishing policy in the corresponding heterogeneous optimal control model is cyclic, and the planned theoretical results should allow to obtain conditions for cyclicity and efficient computation of the optimal solution. Several other fields of application of the results from the two theoretical goals are: multistage dynamic optimization problems involving random events, optimization of advection-reaction processes, health economics, economic geography, etc.

Heterogeneity is an important feature of most of the existing population, economic or social systems. Parameters of heterogeneity of human populations, for example, may be age, preferences, health status, wealth, etc. In a purely economic context, heterogeneity can be brought by the variety of existing technologies or products; since this variety depends on investments in R&D, we encounter heterogeneous models where the domain of heterogeneity is controllable. Such models arise also in the recently developing space economics. The general goal of this project was to enhance the optimization theory for heterogeneous dynamical systems in two directions: (i) for problems where the domain of heterogeneity depends on the decision variables (the control); (ii) for problems involving periodic control. This goal is motivated by numerous new applications in economics and population dynamics, where the existing theories do not provide efficient solution instruments. The investigation carried out within the project, required some non-standard extensions of the optimal control theory and the numerical methods for first order partial differential equations, where the spatial domain (the domain of heterogeneity) is endogenous. In addition, an important result of the project is a new approach for obtaining a complete set of necessary optimality conditions for infinite-horizon problems; it provides an important contribution also to the classical theory of optimal control of ordinary differential systems. Using these theoretical contributions we gained important insights in several economic problems at micro and macro level. In particular, we solved the problem of optimal allocation of investments to capital accumulation and R&D. Moreover, in a general equilibrium model with an endogenously growing product variety we showed that the obsolescence of consumption products is important for boundedness of the gap between savings and investments. We also investigated problems of optimal harvesting of biological resources, where the heterogeneity is either with respect to age/size or with respect to the position in the physical space. We establish that the heterogeneity may be responsible for appearance of cyclical optimal solutions involving recovery periods of the resource. Several complementary investigations are devoted to problems of health investments or optimal education in life-cycle models, optimal emission abatement by heterogeneous agents, markets of emission permits in a vintage capital framework, and optimal immigration age-patterns.